Question

The equilibrium constant \(K_{\mathrm{a}}\) for the reaction $$\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}^{3+}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons{\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{5}}(\mathrm{OH})^{2+}(a q)+\mathrm{H}_{3} \mathrm{O}^{+}(a q)$$ is \(6.0 \times 10^{-3}\) a. Calculate the \(\mathrm{pH}\) of a \(0.10-M\) solution of \(\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}^{3+}\) b. Will a \(1.0-M\) solution of iron(II) nitrate have a higher or lower \(\mathrm{pH}\) than a \(1.0-M\) solution of iron(III) nitrate? Explain.

Short answer

The pH of a 0.10-M solution of Fe(H2O)_6^3+ is approximately 1.61. A 1.0-M solution of iron(III) nitrate will have a lower pH (more acidic) than a 1.0-M solution of iron(II) nitrate due to the stronger acidic nature of iron(III) complexes.

Step-by-step solution

Set up the ICE table

Write down the Initial concentrations (I), the Change (C) and the Equilibrium concentrations (E) of the species involved in the reaction. Initial: [Fe(H2O)6^3+] = 0.10 M, [Fe(H2O)5(OH)]^2+ = 0, [H3O^+] = 0 Change: [Fe(H2O)6^3+] = -x, [Fe(H2O)5(OH)]^2+ = +x, [H3O^+] = +x Equilibrium: [Fe(H2O)6^3+] = 0.10 - x, [Fe(H2O)5(OH)]^2+ = x, [H3O^+] = x #Step 2: Use the Ka expression and solve for x#

Solve for x using Ka expression

Write the Ka expression for the reaction and substitute the equilibrium concentrations. Then, solve the equation for x (concentration of H3O^+ ion). Ka = \(\frac{[Fe(H_2O)_5(OH)]^{2+} [H_3O^+]}{[Fe(H_2O)_6^{3+}]}\) 6.0 × 10^(-3) = \(\frac{x^2}{0.10 - x}\) Since Ka is very small, we can assume that x is also very small compared to 0.10, so the equilibrium concentration of [Fe(H2O)6^3+] can be approximated to 0.10. Now, we can solve the equation for x. x^2 = 6.0 × 10^(-3) × 0.10 = \(6.0 × 10^{(-4)}\) x = √(6.0 × 10^(-4)) = 2.45 × 10^(-2) #Step 3: Calculate pH using H3O^+ concentration#

Calculate pH

Calculate the pH of the solution using the concentration of H3O^+ ions (x). pH = -log([H3O^+]) pH = -log(2.45 × 10^(-2)) pH ≈ 1.61 So, the pH of the 0.10 M solution of Fe(H2O)6^3+ is approximately 1.61. #b. Will a 1.0-M solution of iron(II) nitrate have a higher or lower pH than a 1.0-M solution of iron(III) nitrate? Explain.# #Step 1: Determine the acidic nature of iron(II) and iron(III) complexes#

Determine the acidic nature of complexes

Iron(III) complexes generally have a stronger acidic nature than iron(II) complexes. This is because the higher positive charge on the iron(III) complexes attracts electrons more, making their ligands more prone to being released as H3O+ ions when dissolved in water. #Step 2: Compare pH of the solutions#

Compare the pH values

Since iron(III) complexes have a stronger acidic nature and release more H3O+ ions than iron(II) complexes, it can be inferred that a 1.0-M solution of iron(III) nitrate will have a lower pH (more acidic) compared to a 1.0-M solution of iron(II) nitrate.

Key concepts

pH Calculation

Understanding the pH calculation is crucial in chemistry, particularly when dealing with solutions of acids and bases. The pH of a solution is a measure of its acidity or basicity. It is defined as the negative logarithm (base 10) of the hydrogen ion concentration, which is represented mathematically as:

\( \text{pH} = -\log[H_3O^+] \)

The concentration of hydrogen ions, \( [H_3O^+] \), is directly tied to the properties of the substances dissolved in the solution. For acids, as the concentration of hydrogen ions increases, the pH decreases, indicating a more acidic solution. In the example provided, the pH calculation begins by first determining \( [H_3O^+] \) from the ICE table using the equilibrium constant (Ka) of the hydrolysis reaction of the iron(III)-water complex. Once the concentration of \( H_3O^+ \) is found, the pH is easily calculated. Using the antilogarithm (inverse of logarithm) gives the specific numeric value of the pH, providing insight into the acidity of the solution.

ICE Table Method

The ICE Table method is an invaluable tool for chemists to find the concentrations of particles in equilibrium reactions. ICE stands for Initial, Change, and Equilibrium, reflecting the stages of the reaction. This method is particularly useful when dealing with weak acids, bases, or soluble complex ions.

Here's how one can capitalize on the ICE table:

• Initial: Establish the starting concentrations of reactants and products.
• Change: Indicate the changes (increase or decrease) in concentrations as the reaction reaches equilibrium.
• Equilibrium: Calculate the final concentrations at equilibrium, which are a result of the initial concentrations and changes.

Using the ICE Table simplifies solving for unknown values, such as changes in concentration and the equilibrium constant. It is a straightforward visualization that helps in comprehending shifts in reaction dynamics. In the current problem, an ICE Table facilitates the determination of the hydronium ion concentration, which is the cornerstone for calculating the pH.

Acidic Nature of Complexes

The acidic nature of metal complexes, like those with iron in different oxidation states, arises from their ability to release hydrogen ions into solution. Iron(III) complexes typically possess greater acidity compared to iron(II) complexes. This is attributed to the increased positive charge on the iron(III) ion, which generates a stronger pull on the electrons from its surrounding ligands.

When dissolved in water, the ligands of an iron(III) complex can release hydrogen ions more readily, contributing to a lower pH. This concept helps us predict the behavior of such complexes in aqueous solutions:

• Iron(III) complexes, having a higher positive charge, are more acidic, thus releasing more hydroxide ions into the solution, leading to a lower pH.
• Iron(II) complexes, being less positively charged, are less acidic, thus contributing to a higher pH compared to iron(III) complexes.

This principle allows chemists to infer which solutions will be more or less acidic solely based on the charge and nature of the metal ion in the complex, thereby predicting and explaining the observed pH values in experimental conditions.